Relaxation of star polymers

The scaUng picture of a star can also be used to predict its dynamic relaxation processes. There are at least three qualitative distinct relaxation processes for a star, which occur on different time scales and only weakly couple to each other. While all three of these mechanisms also occur for linear polymers, in a star they are easily separable. 

The fastest process is the shape fluctuations, which can be measured directly by studying the fluctuations of the inertia tensor M which is given by 

Results for C]( t) for stars in a good solvent are presented in Fig. 9.11 for a series of four stars with N = 50. The relaxation times, rd for the elastic modes, which are determined by the slope at long times on this semilog plot are essentially the same. Su et al. found similar results for 3- and 4-arm 

Tg This describes the initial stage of the local relaxation. 

In order to produce an overall shape fluctuation, a density fluctuation must diffuse a distance of the order of the diameter of the star, which is R. The diffusion constant for a semidilute solution is given by divided by the local relaxation time rg. Thus is of order. 

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In particular it has been conjectured that the terminal relaxation of star polymers might be the most sensitive test of the dilution exponent P in Go theta solvents suggest a mean value of nearer 2.3 [32]. A physically reasonable scahng assumption for the density of topological entanglements in a melt of Gaussian chains leads to a value of 7/3 [31]. 

Despite these complications, there are now numerous evidences that the tube model is basically con-ect. The signatory mark that the chain is trapped in a tube is that the chain ends relax first, and the center of the chain remains unrelaxed until relaxation is almost over. Evidence that this occurs has been obtained in experiments with chains whose ends are labeled, either chemically or isotopically (Ylitalo et al. 1990 Russell et al. 1993). These studies show that the rate of relaxation of the chain ends is distinctively faster than the middle of the chain, in quantitative agreement with reptation theory. The special role of chain ends is also shown indirectly in studies of the relaxation of star polymers. Stars are polymers in which several branches radiate from a single branch point. The arms of the star cannot reptate because they are anchored at the branch point (de Gennes 1975). Relaxation must thus occur by the slower process of primitive-path fluctuations, which is found to slow down exponentially with increasing arm molecular weight, in agreement with predictions (Pearson and Helfand 1984). 

The properties of the surplus segment probability p and the effective constraint coordination number z are less well established. It seems possible that p will dep d on polymer species to some extent, since loop projection may be easier for a more locally flexible chain. Weak dependences on concentration and temf rature are likely for the same reason. On the other hand, z characterizes the topology on a fairly large scale and therefore may be virtually a universal constant. Diese however are only some speculations. Values of p and z can be established by various experiments, p from the elastic properties of networks and also from the relaxation of star polymers, z from relative relaxation rates of linear and star molecules in liquids and networks and also from measurements of diffusion rates of stars in linear chain liquids. The adequacy of the... 

D. Richter, B. Stuehn, B. Ewen, and D. Nerger. Collective relaxation of star polymers -a neutron spin-echo study. Phys. Rev. Lett., 58 (1987), 2462-2465. 

Because of the exponential dependence of relaxation time on the potential, the relaxation of star polymers is extremely sensitive to the strength of the potential and therefore to the value of V and of Mg, which sets the value of Z. The correct value of v has been controversial a discussion of this and of non-quadratic corrections to Eq. 9.2 can be found in McLeish [14]. Recent fine-scale simulations using lattice models and real-space pearl necklace models of entangled polymers provide some justification for the quadratic potential and for the value v= 3/2 [15, 16]. As mentioned in Section 6.3, the relationship between and is also open to revision [17]. Hence, adjustments of either or v might be needed to obtain quantitative predictions of the rheology of star polymers. 

Comparing this result with Eq. 9.2, we find that dynamic dilution speeds up relaxation of a star arm by the exponential of an order unity prefactor times a large number Z3. Thus, the degree of acceleration of the relaxation can be truly enormous, i.e., factors of millions or billions. Ball and McLeish point out that inclusion of the dynamic-dilution effect is essential if truly quantitative, or even quahtative, predictions of the relaxation of star polymers are to be obtained. 

Shanbhag, S., Larson, R. G., Takimoto, J., Doi, M. Deviations from dynamic dilution in the terminal relaxation of star polymers. Phys. Rev. Lett. (2001) 87, article no. 195502... 

The best insight into the relaxation behavior of star polymers in dilute solution can be expected if, in addition to the whole star system, different parts of the star are considered separately. This can be achieved easily by neutron scattering techniques on systems where not only the entity of arms, but also single arms, the core or shell parts are labelled by proton deuterium exchange. With respect to the core-shell labelling it is convenient to build up the arms as diblock copolymers of A-B type with protonated or deuterated but otherwise chemically identical A and B blocks. 

The mathematical treatment that arises from the dynamic dilution hypothesis is remarkably simple - and very effective in the cases of star polymers and of path length fluctuation contributions to constraint release in Hnear polymers. The physics is equally appealing all relaxed segments on a timescale rare treated in just the same way they do not contribute to the entanglement network as far as the unrelaxed material is concerned. If the volume fraction of unrelaxed chain material is 0, then on this timescale the entanglement molecular weight is renormalised to Mg/0 or, equivalently, the tube diameter to However, such a... 

For both linear and star polymers, the above-described theories assume the motion of a single molecule in a frozen system. In polymers melts, it has been shown, essentially from the study of binary blends, that a self-consistent treatment of the relaxation is required. This leads to the concepts of "constraint release" whereby a loss of segmental orientation is permitted by the motion of surrounding species. Retraction (for linear and star polymers) as well as reptation may induce constraint release [16,17,18]. In the homopol5mier case, the main effect is to decrease the relaxation times by roughly a factor of 1.5 (xb) or 2 (xq). In the case of star polymers, the factor v is also decreased [15]. These effects are extensively discussed in other chapters of this book especially for binary mixtures. In our work, we have assumed that their influence would be of second order compared to the relaxation processes themselves. However, they may contribute to an unexpected relaxation of parts of macromolecules which are assumed not to be reached by relaxation motions (central parts of linear chains or branch point in star polymers). 

Similarly, in the case of star polymers, the relaxations of the branch point, arm centre and chain end have been differentiated. The influence of branching has been detected by comparing linear chains with arms of star polymers. A slower relaxation is observed in the case of branched species. 

Constraint release is likely to be very important in the relaxation of branched polymer liquids. However, if we ignore that complication, the stress relaxation modulus for a liquid of highly entan ed stars is given simply by Eq. 69 with v replaced by vi. The viscosity and recoverable compliance can then be calculated from Eq. 69 with Eqs. 25 and 26. 

The results of eqns (7.263) and (7.264) are in qualitative agreement with experimental results the viscosity increases steeply because of the exponential factor, and the steady state compliance is pri rtional to M. However, the quantitative agreement is not satisfactory. The observed viscosity is smaller than the calculated one, and the best fit with experiments is obtained only when the numerical coefficient in the exponential of eqn (7.263) is replaced by a smaller number (about 1/2) instead of lS/8. This suggests that relaxation mechanisms other than the contour length fluctuations are important for star polymers. Indeed it has been pointed out that in the case of star polymers the constraint release, and perhaps other tube reorganization processes, are as important as the contour length fluctuation. 

From the considerations presented in the previous section, it is evident that it is desirable to choose MC moves X X such that the relaxation time resulting from a Markov chain of such moves for the configurations of the polymer chains is as small as possible. This is particularly important for dense melts of long (and hence mutually entangled ) polymer chains, where the reptation concepts imply an asymptotic scaling rocN , that is, the relaxation is distinctly slower than for isolated chains (cf. eqns [13]-[16]).The situation would be even worse for dense melts of star polymers (or other branched polymers) where even an exponential scaling (lnr°=N) may result. ... 

Another case where tube contraction is important is the relaxation of branched polymers/as epitomized by star-polymers. Because the branch point is highly immobile, reptation by Brownian motion of an arm as a whole is strongly inhibited. Escape from the tube can only occur by contraction of the primitive path. Any significant contraction has a high free energy (discussed in the exercise above). The time for a fractional contraction i.e. A is... 

We have dealt with the long-time hydrodynamic properties of star polymers in dilute solution in Sections III. A and III.B. The intrinsic viscosity is the sum of products of moduli and relaxation times. In term of frequency (time" )... 

The relaxation dynamics of star polymers has been studied most sue-... 

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